Existing optical systems can present a sequence of two or more images which appear as the viewer changes their angle of regard over a range of less than 90.degree., typically about 45.degree.. The viewer's angle of regard is the angle between a normal to a plane and a viewer's eye. These systems use a device consisting of a regular array of piano-convex cylindrical lens elements with a periodicity that depends on a viewing distance for which the device is designed. For example, at normal viewing distances for reading of 18 inches, the periodicity of the cylindrical elements is preferably 100 elements to the inch or more. Optical system designs of up to 250 elements per inch can be achieved using such systems. Each lens element brings a slice of an underlying printed image into focus. The underlying printed image is composed by a computer program from a desired image sequence, the optics used and viewing geometry. Without the array of lenses, the printed image appears to be a set of stripes that run orthogonally to the axes of the lenses with a multi-image stripe periodicity equal to the lens periodicity but divided into image-specific stripes by the number of images in the set. For example, if the set has eight images and the lenses are at 100 per inch, a periodicity of eight stripes in 0.01 inches are used, one stripe for each image. The viewer sees the intersection of a focused stripe due to the lens and the printed stripe. Technically, two configurations exist, one in which the cylindrical axes are vertical, and one in which the axes are horizontal. In the vertical case, the images are selected to appear at differing depths since the right and left eyes of the viewer have differing angles of regard. The device can be stationary, and, typically, eight images that appear to be at differing depths are superimposed. In the horizontal case, the viewer's angle of regard must be changed. As the device is rotated through a series of angles of regard, a sequence of images is seen. Various effects are possible depending on the design of the stripes. One image can gradually transform into another, termed "morphing". A sequence of images of various stages of an action scene can give a motion-like effect, termed "motion". If the images are unrelated, the viewer will see the unrelated sequence appear, termed "flip". One image can be a magnification of the preceding image and the effect is similar to looking through a zoom lens, termed "zoom".
Existing lens arrays are fabricated by first designing a cutting tool with a desired lens shape, then using this tool to cut the shape of an array of flat lenses into a flat plate or cylinder. This pattern is then transferred to a plastic sheet using conventional methods. The striped image array can be produced by a thermal or piezoelectric ink jet printer that is readily capable of producing 1200 picture elements per inch and laminated to the flat side of the array. When the thickness of the lens array sheet is small enough, the pattern can be reverse printed using a printing press. Reverse printing using a transfer printing press is an option for array resolutions of greater than 60 elements per inch. Another option for directly printing a reverse image on the array is screen printing which is an option for array resolutions of greater than 10 elements per inch up to about 60 elements per inch. The lamination technique is broadly applicable. The printing medium may be plastic or paper. Since the printed array has the same frequency as the lens array, the print that contains the multiple image information must be linearly registered to the lens. The device operation is designed into the image dissection and the lens array. The lens array is typically fabricated from a plastic. Many different materials can be used, for example acrylics, polystyrenes, polycarbonates, polyesters or equivalent materials. The thickness of the finished sheet is related to the periodicity of the lens elements, which effects the lens cross section. The thickness of the array depends on the index of refraction of the plastic material. ##EQU1##
where:
T is the thickness of the sheet, PA1 D is the width of a lens element (in consistent units with T), PA1 n is the index of refraction of the material.
The above formula assumes that the lens cross section is substantially circular. Other cross sections have been proposed. For example, U.S. Pat. No. 5,642,226 suggests a parabolic cross-section. The proportionality relation still holds for the parabolic cross-section, but the constant that must be introduced to make the equation an equality is different.
The criteria for a satisfactory lens design include the ability for the lens to sharply focus on the image plane. In order to provide an unequivocal differentiation from one selection to the next, the uncertainty of the focus at each selected angle of the array must be small enough. For a sequence of eight images, the uncertainty should be about 6% of the width of the lens or less. The range of foci along the dimension orthogonal to the lens axis determines the range of angles available for the image effects. At an angle of regard larger than about 30.degree., each lens focuses on a slice that is under the orthogonal projection of the next lens element on the image plane. However, the selection quality can nonetheless be judged over the .+-.30.degree. range. Within the range of angles used in the design, the foci should stay within such orthogonal projection. The range of useful angles is established using the angle at which the foci cross a line that represents the edge of the projection. For example, in existing designs using acrylic lens material, the line is typically crossed at about 30.degree., yielding a better than 45.degree. viewing range for images. Each image becomes stable with a rotation of about 6.degree.. If the range were very small, the viewer would be challenged to maintain the orientation over an angle much less than 6.degree. or the design would be required to reduce the repertoire of images to fewer than eight. It can be useful to think of this in reverse. That is, determine the angles of regard that correspond to desired selection points. The viewer can easily and unnoticeably vary an angle of regard by a few degrees. Consequently, the set of angles of regard are not required to form a linear series.
Two cross sections have been used in industry, circular and parabolic. The circular cross section has only a single parameter, the radius, which is typically greater than D/2. The choice of a radius determines the constant of proportionality that makes formula [1] into an equality. The reason the radius is greater than D/2 is that adjacent lenses meet at an angle that must be fabricated. Were the radius exactly D/2, the angle would be an impractical 0.degree.. For simplicity of discussion, all dimensions are normalized by dividing by D/2.
After normalization, the viewer's eye is typically 3000 units or more away. From the point of view of an individual element, the change in angles of regard across the element can be neglected. This is not true from the point of view of the array, and the stripe design accounts for this difference. The circular cross section in these normalized units is defined by x.sup.2 +y.sup.2 &gt;1. To analyze the focusing power, a normal to the surface is used to apply Snell's refraction law. The angle of the normal to the circular surface is arctan(y/x).
To apply the solutions disclosed in U.S. Pat. No. 5,642,226, a parabolic cross section is defined by the less familiar equation: y=k(1-x.sup.2), which is presented in the same normalized units. Unfortunately, U.S. Pat. No. 5,642,226 does not teach the range of the parameter k over which the preferred solution exists. The angle of the normal to the parabolic surface is arc tan(1/2kx). The refractive analyses depend on the parameter k. Mathematically, a parabola is a line that is equidistant from a point (e.g., 0, k-k/4) and a line (y=k+k/4) which are commonly termed as the focus and the directrix, respectively. In U.S. Pat. No. 5,642,226, k is positive definite or greater than zero. Because of practical considerations in constructing a lens array, the range of x for these normalized units is -0.8 to +0.8, which allows 10% of the skirts of the lens for molding or extrusion tolerance. Symmetry allows for considering the range of 0.0 to +0.8 for x and positive angles of incidence with respect to a normal to the array. Snell's law is applied assuming that the index of refraction of the plastic material is 1.45 which is in the range for common acrylics. Better designs could be realized with a higher index of refraction materials such as those that are used in plastic lenses for spectacles which are considered cost prohibitive. The following table presents some critical data used in the refractive analyses of the lens array in U.S. Pat. No. 5,642,226.
TABLE 1 Angle of the orthogonal for a circle and example parabolas in degrees x = 0 0.20 0.40 0.80 Circle 90 78.46 66.42 36.87 k = 2 90 51.34 32.01 17.35 k = 1 90 68.20 51.34 32.01 k = 0.5 90 78.69 68.20 51.34
Refractive analysis proceeds as follows: a ray makes an angle with respect to the normal to the surface of the array. From Table 1, the angle the ray makes with respect to the orthogonal is calculated. From Snell'law, the angle of the ray inside the dielectric is calculated (for example, the acrylic) with respect to the orthogonal from Table I. The result of the analysis is that for larger values of k, the thickness of the array is reduced as suggested in U.S. Pat. No. 5,642,226. However, the accuracy of focus is substantially inferior. For large values of k, all hope of presenting eight distinct images to the viewer is lost. As the parameter k is decreased, the focusing power of the lens improves dramatically. However, concomitant with that improvement, the focal plane is no longer improved. Moreover, the angular range is restricted rendering the selection angle range extremely small (much less than 45.degree.).
The numeric results of refractory analysis of the lens array of U.S. Pat. No. 5,642,226 are provided in Table 2. Each parabola and the circular case have been analyzed with incident rays every 5.degree.. The average value of x is reported. If x is in the range of -1 to 0, it is within the orthogonal projection of the lens on the focal plane. An x value of less than -1 signifies that the angular range has been exceeded. Referring to the data for k=2, the range available for the presentation of an image sequence is less than .+-.5.degree. or a total range of less than 10.degree.. Such a small range is unsatisfactory for presenting a reasonable sequence of images. On the basis of this criterion, k must be in the range of 0.5 to 1.0, and the available range for a parabolic cross section is less than 30.degree.. For many applications, this value of k is satisfactory. The rows titled SD (x) are the standard deviations of the values of x taken over a range of ray intersection points. This number measures the ability of the lens to select the correct image at high contrast. An ideal solution would have a 0 standard deviation. On the basis of standard deviation, k =1 can be ruled out because x(10.degree.)-SD (x, 10.degree.)&lt;x(5.degree.)+SD(x, 5.degree.) resulting in focus lines that significantly overlap to reduce contrast. The overall relative thickness of the array is k-opt (y) for parabolic designs and 1-opt (y) for circular designs. Unless the parabolic array is addressed through an aperture that avoids intersecting the array on the skirts of the parabola, the optimum thickness for best foci exceeded the substantially circular design point.
TABLE 2 Results of the analyses of U.S. Pat. No. 5,642,226 Parabola Angle 5.degree. 10.degree. 15.degree. 20.degree. 25.degree. k = 2 x -1.24 -1.55 -1.91 -2.31 -2.75 SD (x) 0.30 1.62 1.94 2.31 2.74 opt y -2.31 -2.55 -2.78 -2.99 -3.15 k = 1 x -0.64 -0.92 -1.23 -1.56 -1.91 SD (x) 0.62 0.80 1.01 1.26 1.54 opt y -2.38 -2.59 -2.77 -2.91 -3.00 k = .5 x -0.47 -0.81 -1.18 -1.55 -1.92 SD (x) 0.30 0.44 0.61 0.81 1.03 opt y -3.66 -3.89 -4.06 -4.15 -4.17 k = .4 x -0.48 -0.86 -1.27 -1.68 -2.08 SD (x) 0.25 0.39 0.55 0.75 0.96 opt y -4.43 -4.67 -4.84 -4.91 -4.90 Circle x -0.08 -0.27 -0.46 -0.65 -0.83 opt y -1.75 -1.82 -1.85 -1.85 -1.80 SD (x) 0.13 0.14 0.14 0.13 0.11
U.S. Pat. No. 5,642,226 offers another solution wherein the transparency is achieved by introducing a parallel concave lens array in the back surface of the primary array. The concave lens is constructed to have a negative focal point equal to the positive focal point of the front-surface lens. This combination allows an image behind the array to be viewed by the observer. In optics, a concave-convex lens is referred to as a meniscus lens. Such a lens can be positive or negative depending on the focal points of the convex and concave surfaces. The normal assumption is that the lens is thin. That is, the thickness of the lens is much less than either focal length.
In U.S. Pat. No. 5,642,226, the requirement is that the convex lens sharply focus the incoming light rays to perform a critical selection function. As a result, the concave portion of the array perforce substantially focuses light at the intended "see through" angle before striking the concave lens. This combination does not act as a thin, zero-magnification meniscus lens. The negative lens does allow the transparency objective to be realized. However, an undistorted view of the image positioned behind the array is available only on a precisely placed plane. Three-dimensional objects are unfortunately distorted. The position of the plane bears a fixed relationship to the thickness of the primary array. If the primary array is on the order of 0.015 inches, thick, the rear image must be less than 0.1 inches from the rear of the primary array for best foci.
U.S. Pat. No. 5,642,226 further suggests that a lens system can be realized using diffraction rather than refraction for the objective of an anisotropic lens array that acts as a window over a set of angles of regard and produces a second image or sequence of images over another set of angles of regard. The fundamental principle is that a transparent object when illuminated by a coherent light produces a spatial distribution of light amplitudes that characterize the object. If a cylindrical lens is so illuminated, the lens gives rise to a set of refracted spherical waves, one corresponding to each point of the lens. Since the illumination is coherent, each wave of the set of refracted spherical waves is mutually coherent and produces an interference or diffraction pattern determined by the relative refraction of every point in the lens. The pattern is characteristic of the lens. Generally, the pattern is captured in a photosensitive emulsion. Since photosensitive emulsions are sensitive to light intensity and not the amplitude variations produced by the diffraction pattern, the emulsion is illuminated with a reference beam that is coherent with the beam that illuminates the lens in addition to the light that characterizes the object. This is referred to as a phase-reference hologram and represents the most effective way to realize such a lens. Diffraction-based lenses such as the one described in U.S. Pat. No. 5,642,226 and earlier devices such as Fresnel zone-plate lenses are rarely seen in optical devices for managing incoherent white light since the quality of performance of the diffraction pattern depends strongly on the wavelength of light.
U.S. Pat. No. 5,642,226 further suggests that a lens system can be realized using the Fresnel lens technique when designing lenses for the objective of an anisotropic lens array that acts as a window over a set of angles of regard and produces a second image or sequence of images over another set of angles of regard. The imaging of lenses depends primarily on the surface curvatures of the surfaces of the dielectric rather than the thickness of the dielectric material. The focusing effect of a normal lens can be obtained in an optical element if the surfaces are divided into small elements and these elements are brought together in a common plane that is normal to the optical axis. This is referred to as the Fresnel lens technique. Such lenses are termed "Fresnel lenses". Unfortunately, breaking up the aperture of a lens into small zones destroys the continuity of the wave front. Therefore, such lenses do not provide high performance or clarity of image. The technique is useful where high performance is not necessary and/or where the thickness of the lens is paramount.
A high zone count Fresnel lens and a Fresnel lens that features the actual surface shape of the lens being simulated as opposed to the more easily fabricated linear approximation will be required to apply the Fresnel lens technique to situations where the image repertoire is in the typical range of eight. For directly viewed anisotropic lens arrays where the viewing distance is approximately 18 inches, the preferred lens width is 0.01 inches or less. For a circular cross section lens for this viewing distance, the overall thickness is approximately 0.015 inches. The maximum thickness saving using the Fresnel lens technique is less than 0.05 inches. The saving approaches 0.05 inches as a limit as the number of zones increases. As the number of zones increases, the precision of the die from which such arrays are fabricated perforce increases as does the fabrication cost. For ten zones, the die precision increases at least threefold. To effectively select from a repertoire of eight images, each image's angular range should be sampled by at least two, preferably three zones. This implies that at least sixteen, preferably more zones are required even further exacerbating die complexity and molding costs. Solutions based on the Fresnel technique are best applied to a long viewing distance array with limited image repertoires. For example, for a 300-foot viewing distance, the preferred lens width is 2 inches and the array thickness is approximately 3 inches. Saving 30% of the material given, a 300-foot design point may justify the Fresnel technique. In such billboard applications, a lens' array can be assembled in situ by abutting array segments that are several feet wide and tens of feet long. Unfortunately, for most applications, the Fresnel technique is contraindicated.
U.S. Pat. No. 5,642,226 further suggests using a transparent dielectric, such as acrylic, with the viewer side printed with opaque stripes and the far side printed material sampled by stripes of the pitch, aligned with the viewer side stripes. Over one range of angles of regard, the viewer observes the clear portion of the back of the device interstitial to the opaque front side stripes. Over another range of angle of regard, the viewer observes the printed image on the back side of the device interstitial to the opaque front side stripes. This simple device is suitable in the situation where there is a desire to see a single image or a window depending on the angle of regard. Provided the spacing of the opaque stripes are small compared to the viewing distance (1:1000 or better) and the thickness and index of refraction of the plastic sheet are appropriately chosen so that the refraction limited angular range is not a concern. Unfortunately, this device cannot encompass a sequence of images that appear as the angle of regard changes.
U.S. Pat. No. 4,541,727 introduces voids into an image affixed to the backside of a standard lens array. The array can permit the viewer to see a first predetermined image printed on the backside of the array or, over another range of angles of regard, see an object behind the array. If the image behind the array were a watch (the example used in U.S. Pat. No. 4,541,727), at several times of the day the hands of the clock are positioned at right angles, for example, 3 o'clock. One hand aligns with the axes of the cylindrical lenses of the array and becomes visible. The other hand, being at a right angle with the first hand, is aligned at a right angle with respect to the array. This hand is out of focus since the foci are perforce on the backside of the array. Further, this hand is optically segmented by the array of lenses. Experimentally, this hand is so distorted in view as to disappear entirely.
U.S. Pat. No. 5,644,431 discloses a first structure that is substantially similar to one of the structures in U.S. Pat. No. 5,642,226 and previously discussed hereinabove. U.S. Pat. No. 5,644,431 acknowledges the distortion of see-through image and designs a focal length of a negative lens to assure that the emerging rays are substantially parallel. This solves the problem found in U.S. Pat. No. 5,642,226 of the requirement for a precise and generally very small distance of the see-through image from the back of the array. The assertion is that the distortion is tolerable since the distortion amounts to graininess in one dimension of the object behind the array. The array will demagnify the see-through image in one dimension but not the other. U.S. Pat. No. 5,644,431 also discloses an array of spherical micro lenses. The back side of the array contains an array of spherical micro lenses in one to one correspondence with the front side. The lenses on both sides of the array are positive or convex. U.S. Pat. No. 5,644,431 suggests that two plano-convex array sheets should be fabricated and then aligned and bonded together to form the final array. The back side is no longer suitable for imprinting images due to the back side lenses. Further, achieving see-through with the array disclosed by U.S. Pat. No. 5, 644,431 is undermined because the back side is out of focus with respect to the front side lenses.
What is therefore needed is a high performance lenticular system that provides an image sequence as a viewer's angle of regard changes. Further needed is a lenticular system having a display of a repertoire of two or more images from a viewer side while simultaneously permitting viewing through the display at a pre-determined angle of regard. Further needed is a lenticular system that may be manufactured using non-complex dies and inexpensively molded and that provides an image sequence using a micro lens array wherein a printed image may be laminated or otherwise affixed to a back side of the array.